Paper by Erik D. Demaine

The 15 puzzle is a classic reconfiguration puzzle with fifteen uniquely
labeled unit squares within a 4 × 4 board in which the goal is
to slide the squares (without ever overlapping) into a target configuration.
By generalizing the puzzle to an n × n board
with n2 − 1 squares, we can study the
computational complexity of problems related to the puzzle; in particular, we
consider the problem of determining whether a given end configuration can be
reached from a given start configuration via at most a given number of moves.
This problem was shown NP-complete in [1]. We provide an alternative simpler
proof of this fact by reduction from the rectilinear Steiner tree problem.